A Weak Reverse Holder Inequality for Caloric Measure

TL;DR

This paper establishes a weak reverse Holder inequality for caloric measure under parabolic Ahlfors-David regularity, linking it to solvability of the Dirichlet problem with L^p data in open sets.

Contribution

It introduces a new criterion for caloric measure to satisfy a weak reverse Holder inequality based on parabolic regularity conditions.

Findings

Weak reverse Holder inequality holds under parabolic Ahlfors-David regularity.

Weak reverse Holder estimate is equivalent to Dirichlet problem solvability with L^p data.

Results extend non-doubling harmonic measure theory to caloric measure.

Abstract

Following a result of Bennewitz-Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set $\Omega$, assuming as a background hypothesis only that the essential boundary of $\Omega$ satisfies an appropriate parabolic version of Ahlfors-David regularity (which entails some backwards in time thickness). We also show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with "lateral" data in $L^p$, for some $p<\infty$, in this setting.